A spherical capacitor has an inner sphere of radius 12 cm and an outer sphere of radius 13 cm. The outer sphere is earthed and the inner sphere is given a charge of `2.5 muC`. The space between the concentric spheres is filled with a liquid of dielectric constant 32.
(a) Determine the capacitance of the capacitor. (b) What is the potential of the inner sphere ?
(c) Compare the capacitance of this capacitor with that of an isolated sphere of radius 12 cm.Explain why the later is much smaller ?

Explanation

(a) The capacitance of the capacitor with outer sphere is earthed and inner sphere is given charge can be calculated as:
`C=K4pi epsilon_(0)((r_(1)r_(2))/(r_(1)-r_(2)))`, where `r_(1) and r_(2)` are radii of the outer and inner spheres respectively.
`therefore C=(32)/(9xx10^(9))xx(0.12xx0.13)/((0.13-0.12))=5.547xx10^(-9)F`
(b) Potential of the inner sphere is equal to the potential difference of the capacitor, as potential of the outer sphere is zero.
`therefore V=(2.5xx10^(-6))/(5.547xx10^(-9))=4.51xx10^(2)V`
(c ) Capacitance for an isolated sphere of radius R can be calculated as:
`C.=4pi epsilon_(0)R=(12xx10^(-2))/(9xx10^(9))=1.33xx10^(-11)F`
`(C )/(C.)=(5.54xx10^(-9))/(1.33xx10^(-11))=4.166xx10^(2)=416`
The capacitance of the spherical capacitor is 416 times the capacitance of an isolated sphere.
Presence of earthed conductor near a charged conductor increases capacitance.